How can I animate these planes for Unity game Assets?
Is it your plan to just rotate arms and legs as if It where 2D pixel art?

Use the timeline (the meter with the frame numbers and the green line indicating current position, at the bottom) to go to say frame 25. Now rotate the legs to pose the character as if it were walking, press key "i" to insert a new keyframe, in the menu that just appeared click on "LocRotScale", now drag the green line in the time line from 25 to 0 and from 0 to 25 again, watch how your character walks. Apply this to do more complex movements.

The good thing is that you only need to pose the character in the initial position, then advance x number of frames (depending on how fast you want the movement to be) and pose it in the final position and insert keyframe there. The frames in between will be extrapolated by Blender. You only need to insert more keyframes in the middle of a movement to do more complex movements.

Also, you probably want linear extrapolation of keyframes: How do I make animations a steady speed?

Yes, you can use the Action Editor to save each animation as actions but... I would not trouble myself into do so, I would just save all animations as a continuous very long time line. If you will export as a 2D spritesheet that is precisely the more practical to do. If you will use it as 3D assets maybe you need the actions in order to trigger them by action name from the game engine code.

I don't sure about the state of .blend files support in Unity, I don't know If It will import the actions at all.

No, you don't need to merge all planes into one object. Actually, by having them as separated objects, you can insert keyframes for each plane individually and move them independently. If you join everything into a single object, you will have to use an armature to animate legs and arms (because I think that changes done in Edit Mode to the mesh are globally applied across all keyframes making it impossible to create keyframes by rotating things in Edit Mode. While poses done with armature can be keyframed).